Fluctuations in multiplicative systems with jumps

Fluctuation properties of the Langevin equation including a multiplicative, power-law noise and a quadratic potential are discussed. The noise has the Levy stable distribution. If this distribution is truncated, the covariance can be derived in the limit of large time; it falls exponentially. Covariance in the stable case, studied for the Cauchy distribution, exhibits a weakly stretched exponential shape and can be approximated by the simple exponential. The dependence of that function on system parameters is determined. Then we consider a dynamics which involves the above process and obey the generalised Langevin equation, the same as for Gaussian case. The resulting distributions possess power-law tails - that fall similarly to those for the driving noise - whereas central parts can assume the Gaussian shape. Moreover, a process with the covariance 1/t at large time is constructed and the corresponding dynamical equation solved. Diffusion properties of systems for both covariances are discussed.